Asymptotic eigenvalue distribution of block Toeplitz matrices and application to blind SIMO channel identification

Abstract

Abstract-Szegö's theorem states that the asymptotic behavior of the eigenvalues of a Hermitian Toeplitz matrix is linked to the Fourier transform of its entries. This result was later extended to block Toeplitz matrices, i.e., covariance matrices of multivariate stationary processes. The present work gives a new proof of Szegö's theorem applied to block Toeplitz matrices. We focus on a particular class of Toeplitz matrices, those corresponding to covariance matrices of single-input multiple-output (SIMO) channels. They satisfy some factorization properties that lead to a simpler form of Szegö's theorem and allow one to deduce results on the asymptotic behavior of the lowest nonzero eigenvalue for which an upper bound is developed and expressed in terms of the subchannels frequency responses. This bound is interpreted in the context of blind channel identification using second-order algorithms, and more particularly in the case of band-limited channels.Index Terms-Asymptotic eigenvalue distribution, band-limited channels, blind identification, block Toeplitz matrices, multivariate processes, second-order statistics algorithms.

Information theoretic capacity of Gaussian cellular multiple-access MIMO fading channel

“…In this case however the matrix is a block circulant matrix and Szego's theorem cannot be directly applied. However using the results in [9] the block circulant matrix in (9) can be transformed to an equivalent matrix, from an eigenvalue point of view, thus allowing us to obtain an asymptotic expression of the maximum capacity for any finite number of users K.…”

“…Now taking the two dimensional Fourier transform of each T(τ u,v ) we can find the asymptotically equivalent matrix, from an eigenvalue point of view, of the matrix in (9). This is the circulant n r × n r matrix: …”

“…Now starting from (4) and applying theorem 3 of [9] we can find a closed form for the uplink capacity of a MIMO cellular system.…”

“…Following a similar methodology as the one presented in [9] for Block Toeplitz matrices we consider the associated matrix:…”

“…In section II we present the channel model and the main input versus output equation. In section III the channel matrix is mathematically formulated while in section IV based on the work presented in [9] for large random block circulant matrices the cell capacity under any fading environment is found. In section V the analytical and simulation results are presented.…”